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Transformation between cartesian and
spherical tensors
A.J. Stone
a
a
University Chemical Laboratory, Lensfield Road, Cambridge, CB2 1EW,
England
Version of record first published: 22 Aug 2006
To cite this article: A.J. Stone (1975): Transformation between cartesian and spherical tensors, Molecular
Physics: An International Journal at the Interface Between Chemistry and Physics, 29:5, 1461-1471
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MOLECULARPHYSICS, 1975, VOL. 29, No. 5, 1461-1471
Transformation b e t w e e n cartesian and spherical tensors
Downloaded by [North Carolina State University] at 05:56 03 September 2012
by A. J. S T O N E
University Chemical Laboratory, Lensfield Road,
Cambridge CB2 1EW, England
(Received 13 May 1974; revision received 2 October 1974)
A standard unitary transformation is proposed for interconversion between
cartesian tensors and spherical tensors, and between expressions including
such tensors. Methods for manipulating the transformation coefficients are
described, and the effects of symmetry with respect to permutation of
cartesian tensor suffices are discussed. As an example, the method is used
to derive the angle-dependence of the circular intensity differential for
Rayleigh scattering from a dimer.
1. INTRODUCTION
Calculations involving tensor quantities are more conveniently expressed
sometimes in terms of cartesian tensors, and sometimes in terms of spherical
tensors. It is therefore advantageous to have a well-defined procedure for
transforming between the two, so that properties expressed in one form (' the
tensor is symmetric with respect to permutations of subscripts ', ' all spherical
components with m # 0 vanish ') can be readily expressed in the other. In spite
of this, no well-defined transformation seems to have been proposed, and the
purpose of this paper is to suggest a suitable transformation and to investigate
some of its properties.
Using the symbol ~7to distinguish, if necessary, between spherical components
of the same rank derived from a given cartesian tensor, we can define a transformation (,/j; ra I ~la~. 99an) and its inverse by
T .......... = X Tnj;m<~j;m] (xlaa...ogn>
~jm
(1.1)
and
T~j;,n= ~
T .......... (ala2... a~l qj;m).
(1.2)
61...~t
If we require that the transformation be unitary, then
<ala~...a~ 1~/j;m> = (,~j;m I o~la~... ~ > * .
(1.3)
To find the transformation, we note that any tensor T ........ ~ of rank n
transforms under rotations in the same way as the tensor ,z/~ B~ ... Z~n, which
may be called a polyadic by analogy with the term ' dyadic ' used for a product
1462
A.J.
Stone
of two vectors. Each vector in this polyadic can be transformed to contrastandard [2] spherical form:
A1;m = ~.z/~(ct [ 1 ;m),
where the unitary matrix of coefficients (al 1 ; m ) is
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Ill)
t10)
t1-1)
(x I [--@liK
0
@89
~Yll ~
iK0
@OK);
0@IK
(1.4)
here and throughout this paper, Condon and Shortley's phase convention [1] is
obtained if K is set to - i, and Fano and Racah's [2] if K-- 1.
The spherical vectors can now be coupled together, so that, for example,
(AB)lJ,;m= Z Z ( ~ [ 1 ; m ' ) A ~ , ( f l [ 1 ; m " ) B ~ ( l l m ' m ' ] j 2 r n )
= ~A~B~(o~[31 1/2;m),
aS
(ABC)lhh;m= ~., (AB)hm,Clm.(julm'm"[jam)
(1.5)
r/t'm'"
X
lj2;m')C~,(%ll;m")(j21m'm"lJam)
X A~,B~,(%~
-- Z A~,B~,C~,(ala,a3l
lj2Ja;m>9
(1.6)
~t~l~a
(Here and subsequently,
(~t~2llje;m)
(jtj2mFnz [jm) is a Clebsch-Gordan coefficient.) Thus
= ? n tY~
~ pp
(~,l]l;m')(o,2ll;m")(llm'm"lj2m),
lj~jz;m)= m'm"
~ (~a~ I lj2;m') (az[ 1;m ") (j21m'm"lj3m)
(alaaa3]
(1.7)
(1.8)
and in general
.
1
= ~ (~,~...~,,_~[lj~...j,_l;m')(~,ll;m')(j,_dm'm"lj,~m).
(1.9)
T h e order in which the vectors of the polyadic are coupled is arbitrary, but
some choice has to be made, and the one made here seems as good as any. It is
necessary to specify the intermediate quantum numbers J2...jn-1, which
correspond collectively to the symbol r/used above, since they distinguish different
spherical components with the same j , . It will be convenient to include jj = 1
in the notation (there being then one j for each suffix of the cartesian tensor)
and to remember that there is also a notional j 0 = 0 corresponding to the
transformation of a scalar:
A0;o=A(I ; 0 ) = A .
We may denote the set of 2 j n + l coefficients (oq...cx,]lj2...jn;ra) by
( % . . . ~ I 1J2 -. .in).
The coefficients will be called cartesian-spherical transformation coefficients, or more briefly CS coefficients.
1463
Cartesian and spherical tensors
2. PROPERTIES OF CARTESIAN-SPHERICAL TRANSFORMATION
COEFFICIENTS
Since the vector coupling transformation is unitary, the whole transformation
is unitary as required. A further property which can be readily proved
inductively from (1.9) is that
( % ' " "%*1 l j = . . . j , ; - m ) = K2"(- 1)&-m<cq... %] l j = . . . j , ; m)*.
(2.1)
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The transformation coefficients for n = 2 are given in table 1. For n = 3 the
matrix of coefficients is 27 x 27, and direct use of the explicit transformation
becomes impractical. Fortunately it is not necessary, as will be shown.
Common
factor
110;0>
111;0)
3-1/2K2
2-1/2K2
<xx[
(xyf
(xzl
1
(yy]
(yz]
(~xl
(zy]
1
111;1)
89
112;0>
[12;1)
6-1/'~K2
89
1
89
-1
-i
-i
1
(yx]
112;2>
1
i
-i
1
1
i
i
1
i
-1
-i
1
-2
T a b l e 1. C a r t e s i a n - s p h e r i c a l t r a n s f o r m a t i o n coefficients for s e c o n d - r a n k cartesian tensors.
T h e c o m m o n factor multiplies all entries in the c o l u m n . Coefficients n o t t a b u l a t e d
can b e o b t a i n e d u s i n g e q u a t i o n (2.1). K = I to o b t a i n F a n o a n d R a c a h ' s p h a s e
c o n v e n t i o n , or - i to o b t a i n C o n d o n a n d Shortley's.
In another paper [3] a method is described for manipulating expressions
involving C S transformation coefficients using an extension of the graphical
technique given by Brink and Satchler [4], this being itself a modification of the
technique proposed by Yutsis et al. [5]. With the help of the graphical technique
the following results can be obtained. The general formulae are fairly complicated, but are not unduly difficult to handle in practice, since only tensors of
fairly low cartesian rank are normally involved.
2.1. Contraction of adjacent suffixes
(%'"
%1 11"'" lr,P )~i~i+ 1
= K$(~Xl " ' " Oli--1Ot/+2''" cXrl / 1 ' ' "
Thus if T~I .... i-1~i+2 .... r=R~l
....
li--lli+2 "'" r
1 )/(21i_ ~ + 1 )]1/2.
x ( - 1)1i-l-~+18tr
(2.2)
r3~i~i+l, then T has spherical components
T11... h;mt = ~R~I .... r$~i,i+l ( ~
9 9 ~176
= XRll...lr;P r(ll"'" Ir ;Pr ] ~
• (0~1""" C~i-10~i+2"""
Ik...3;
9 9 9 O~r
~ )~aiai+l
"Jr;mr),
m,)
1464
A . J . Stone
and substitution of (2.2), together with the unitarity of the transformation,
leads to
Tll""li--lli+2""lr=K2 ~ Rll 'r~li+lli--l(--1)li-1-/i+l \2li(2li+11+1,]
~ 1/2 .
" ' "
(2.3)
li,li + 1
For example, if T = R ~ then
T = x z ~ R,1,23~20( - 1 )0-/1+1(211
+
1 )1/2
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(remembering that l0 = 0) or
T = R ~ = K~v/3Rxo;0.
(2.4)
Although this example tells us nothing new we shall find that a similar procedure
is helpful in dealing with rank 3 and 4 cartesian tensors.
2.2. Permutation o[ adjacent suffixes
Exchange of adjacent suffixes in a cartesian tensor changes the order of
coupling of the vectors of the polyadie, and introduces a 6 - j symbol : If
T~1~2 .... , = (i, i + 1)R~I .... i~i+l .... t
R a 1 . . . ai_lai+laiai+2.., at,
then
Tll" ..l=
~ ( --
l
1,-+,[ nll"" ./i-l//i+1...It'
)'i-1[(2li + 1) ( 2 / + 1)11/3
or in terms of the transformation coefficient,
(i,i+
1) ( a l . . . a,l/1..,
l,;pt}
= Z(-)"-q(2/~+
{
]
, 1)(2f+ 1a~])/
x ( a x . . . at I/1...
~+11i--1~}
li-a~li+a.., l, ;p, }.
(2.6)
Any permutation can be expressed as a product of transpositions of adjacent
elements, so all the permutation properties can be derived from (2.6). Also
contraction with respect to any pair of suffixes may be achieved by use of (2.6)
and (2.3).
2.3.
Inner products
(i)
R-1.-. arSal -.. ar = ~ (K2)r-~rRtl ..-'r ~
(2.7)
l~...1r
(ii) If T~x . . . . t = R~I .... ,+sS"t+l .... ,+s' then the spherical components of T
are given in terms of those of R and S by
Zll...lt,pt=K2s
Z
k I ... k s
Z
//+l--.//+s
(RI1...II+s X Skl...ks)~Pl
x ( - 1)-ks~at+s -/, f i {[(2k.+ 1)(2l,+o + 1)] a/z
a=l
x W(ko_xll, l,+o; kfl,+o_l)}.
(2.8)
Cartesian and spherical tensors
1465
Some particular cases of this result are the following: If To=R~BS B then the
spherical components of T are
Tz;," = Kz]~( _ )tz
(RI/2 x Sx)z,, ;
(2.9)
12
if T~a=R~orS ~ then
(21,~ + 1 ~1/2
_ )'3-'2+z \ ~ /
(R~,2,3 x Sz),, m ;
T1,2;. = tr
(2.10)
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if T~=R~t~Sz~,, then
Tz;m= ~~ (--)k2[(21~+ 1) (21a+ 1) (2ka+ 1)/3] x/~
k212l 3
X
(Rll2l3 X Slk2)lm ;
12
(2.11 )
and ifT.o=R~a~,sS~,8, then
Tlt2;m = X ( -
)k2[(213+ 1) (2/4+ 1) (2k2+ 1)/(212+ 1)] x/2
kzlsl~
13 14}(
112/314 X 8 1 k 2 ) 1 2 m .
(2.12)
2.4. Outer products
If T~I .... rOl''" as = Rat.-. 0~tS~l... Bs' then
k I ... k s
s--1
• I I {[(2jr+o+1) (2ko+ 1 + 1)]ll2W(k,,ljrjr+o+l;ko+lL+o)} (2.13)
o=1
where t = r + s.
In the above formulae,
(R..., xS...k~)m=_
~, (lrksm'm"lym)R...ta,,S...k,m,,
(2.14)
and W ( . . . ) is a Racah coefficient.
3. CIRCULAR INTENSITY DIFFERENTIAL IN RAYLEIGH SCATTERING
The Rayleigh scattering intensity for an optically active molecule is slightly
different for right and left circularly polarized light [6]. If the molecule can be
treated as being composed of two identical optically inactive units, each having a
threefold or higher symmetry axis, then the circular intensity differential A, has
the form [6]
Az = (2co/c)%~ yR~% 8mo~,8(2)/(3%a%a - %~%B),
(3.1)
where c%o('o is the polarizability of unit n, %~ = %(1)+ %e(2), R~ is the position
of unit 2 relative to unit 1, and co is the angular frequency of the light. Let us
determine the dependence of A~ on the relative orientation of the two units.
We will use the Fano and Racah phase convention (K = 1).
1466
A . J . Stone
T o evaluate the denominator in spherical form we can use (2.7) and (2.4):
3%~%B - a~a%, = 3 ~ (
-
)12(0~1/2.
3%o. 0iX0= 3(% 2 . 0t12),
0/1/2) --
(3.2)
12
where we have used the fact that the rank 1 spherical component of a symmetric
tensor vanishes.
If the principal axes of unit n are oriented at Euler angles (%~,~Yn),then
O~12;q(n) =
~o~12~q~(O)~2qq.(
OLn[~n~/n)* = -- oL12;o(O)C2q(f~nOLn),
(3.3)
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q
where O~12;q(0) a r e the components in the principal axes, all of which vanish except
%~;o~~ when there is a symmetry axis of order 3 or more. Using the spherical
harmonic addition theorem, we then find
3 % o ~ o - %~%~ = 3 [o~12.0(0)]2(3 COS2 ~ + 1),
(3.4)
where f~ is the angle between the symmetry axes of the units.
The numerator is more complicated. Define the tensor Aso = % ~~l)c~a0(~), and
consider the inner product %ByAo,. This can be evaluated using (2.11) and the
fact, which will be demonstrated below, that all spherical components of %or
vanish except for the scalar qlo:o= - ~/6. We get
(e.B~,AB~,)lm=~.(_
)k,[2k2 + l]l/2{11 k1 2 01} (gllO
~.
X .~41kt)l m
1 10}(-X/6)Anm
= -- ~r
and
(3.5)
%,~Ao~R~=
- ~/2A n . R1,
(3.6)
since the spherical scalar product of two vectors coincides with the cartesian
scalar product. T o determine Alx;m we note that A~o is a contraction of
c%v(1)a6B(2), so that using (2.3):
Axl=K2~(a(1)~
1 / 2 1 , 1 , ~
= K2V/-,~ [(o~(1)o~(2))1o11 - ~/3(o~(1)~(2))1111 + V5(o~(1)o~(2))1211],
(3.7)
Finally the spherical components of the outer product c~ m%,(2) can be
obtained in terms of those for the individual %0 (n) by means of equation (2.13).
Remembering that %1;m = 0, since c%~ is symmetric, one finds
A l l : m = -- I V 5 ( ~ 1 2 ( 1 ) X
~;12(2))1m,
(3.8)
and it is then straightforward to show that
qq',,,\q
q
Otl2;q(1)Otl2;q,(2)R1;r..
RClm(O,~)we find
~ (~ 2,1m)C2~(fl,%)C2q,(~%)Cl,(Oqh).
(3.9)
Using (3.3) again, and writing R1;m =
A,
x/3OoaR
3c(3 cos ~ f~ + 1) ~'m
q
(3.10)
Cartesian and spherical tensors
1467
It we take the polar axis along R, so that 0 = ~b= 0, and consider the relatively
symmetrical situation where fll = r r - fl~ = fl, ~1 = - ~ = ~b/2, the final result is
(oR
A~ = - 2c( 3c~
4.
1)
SYMMETRY WITH
[sin4flsin2~b+2cos2fisin2flsin~b].
(3.11)
RESPECT TO P E R M U T A T I O N OF
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CARTESIAN SUFFIXES
Most tensors of physical interest have a definite symmetry under permutation
of their suffixes, which reduces the number of independent components. T h u s a
general cartesian tensor of rank 2 reduces to spherical tensors of ranks 0, 1 and 2,
but if the tensor is symmetric all the rank 1 spherical components vanish.
We can project out of any tensor the part which has a particular symmetry
under permutations :
T, I
""(~l
r~=
clvz~ry~,(p_t)pT, 1
tf P
(4.1)
."at'
where ~ r
is an element of the representation matrix for representation F
(of dimension dl. ) of the symmetric group eT~twhose elements P permute the %
The corresponding spherical tensors are given by
Tnr,lt;ptoc ~
al...a t
< a l . . . OLt] 1~. ..
lt;pt ) ~.t Z ~v,~(P-1)PT~x .... ,
9 p
E T~, .... t ~ r ~ ( P - i ) P - l < a l . . . ~ t l l x . . . I t ; p t >
(4.2)
C~l...~t
where in the last line the permutation p - 1 appears as a result of relabelling the
d u m m y suffixes % . . . . . at. The proportionality sign is used because the
projection does not yield a unitary transformation without renormalizing. The
labels 11...lt_ 1 have been dropped because a permutation causes a recoupling
of the basis vectors As, , B~,, etc., and therefore mixes tensors with different
intermediate quantum numbers, as can be seen in equation (2.5). There may
be more than one spherical component with the same Fyltpt, and the symbol 7/is
then required to distinguish between them. Note that equation (4.2)
is independent of Pt, so that any symmetry properties are common to a whole
tensorial set.
Corresponding to (4.2) we can define a new transformation coefficient:
(OL1...
art
~ F f l ; p > = c(vFfl) Z ~ r y ~ ( P ) P < a i . . . a t [ 11...
P
It-ll;P>,
(4.3)
where the constant c07I'yl ) will be chosen to make the transformation unitary.
From equation (2.13) two special cases can be identified. First, for the
permutation (12) we have i = 1, so li_1=0 , li=[= 1. The 6 - j symbol simplifies
and (2.13) reduces to :
(12)<oq...o~tllll2...lt;pt>=(-)',<%...atllll2...lt;pt).
(4.4)
Second, for the transformation coefficient ( % . . . at1123.., t ;Pt> appropriate for
the highest-rank spherical tensor associated with the given cartesian tensor, we
1468
have
A.J. Stone
li=[=i
and
(i'i+l)(~l""~tll2""t;Pt)=(2i+l)
i+1 i ( % " ' ~
(4.5)
The 6 - j symbol can be evaluated using the extended symmetry discovered by
Regge [7] to give
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(i,i+l)(%...at[
12...t;p,)=(2i+l)
{O2 ii l}(%...~,'12...t;p,)
= (Oil... Otll 12... t;pt ).
(4.6)
Thus the highest-rank spherical component of a given cartesian tensor is always
totally symmetric under permutation of cartesian suffixes.
5. THIRD-RANK CARTESIANTENSORS
The general rank 3 cartesian tensor T ...... transforms into seven spherical
tensors Tloa, Tllo, Tin, 7'112, T121, 7'122 and T12a. We have seen that T123 is
totally symmetric; the scalar (isotropic) component Tll0; 0 is easily shown to be
antisymmetric using equation (2.13)--apparently a new if rather heavy-handed
demonstration of this well-known property. Explicit calculation shows that
(~fl~'IllO;O)=--Ka/~/6%Bv, or equivalently that ~110;o=-K3~v/6 , ~lj,l,=0
otherwise.
For the rank 1 and rank 2 tensors we obtain degenerate representations.
If we use the abbreviation ~,l,13----(aflV] ld213;Pa)we find from (2.13) that
(12)(lol 111121)---(lol u 1 1 2 1 ) [ 1
0
~0 - 1
0
0\
O)
(5.1)
1~/5 )
(23)(101111121)=(101121121)(
1
~/89
1/6
x/89
1
-v'(5/12).
~V5 - V(5/12)
(5.2)
The representation matrices for the other elements of o503 can be derived from
these. The representations of oSP3 are given in standard orthogonal form by
Hamermesh [8]; they are the symmetric representation (denoted [3] or [111]),
the antisymmetric representation ([13] or [321]) and the degenerate representation
[2,1] with components [211] and [121]. It is easily shown by standard methods
that the representation spanned by the rank 1 tensors reduces to [3] + [2,1].
The symmetric component is
(~fly] [11111 ) = 89
(c~flV[ 101 ) + ~(~fl~l 121 ),
(5.3)
and the components of [2,1] are
(afl~'l [21111 ) = -~(~flYl
[12111)=
101)+.~C5(a/3~'1121),}
1111).
There is an arbitrary choice of phase in (5.3) and another in (5.4).
(5.4)
Cartesian and spherical tensors
1469
For the rank 2 tensors we find
(12)(u2 1 2 2 ) = ( n 2 1~2) ( - 1
01)
0
(23)(112 1~2)= (112 lZ2) (89189 1V'3~_89
]
(5.5)
so that they form the components of another two-dimensional representation:
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<nflV[ [21112) = <~/3y [ 122),~
<~flTI [12112)
<~fly[ l 1 2 ) , J
(5.6)
where again there is an arbitrary choice of phase.
For most purposes one is interested only in symmetric components. Table 2
gives the transformation coefficients <~x%%[ 123 ), but for the symmetric rank 1
case a simpler form than (5.3) is available.
Common
factor
1123;0)
1123;1)
[123;2)
1123;3)
10-1]2K3
120-*/~K 3
12-1/~K s
8-x/21 s
-3i
<XXX
(3)
(3)
(3)
(6)
(3)
<xxy
<xxz
<xyy
<xyz
<xzz
i
1
-1
-i
--1
--/
1
4i
3
<YYY
(3) <yy~
(3) <yzzl
1
i
-4
2i
-
Table 2. Values of (otfly[123m) for m~>0. The common factor multiplies all entries
in the column. Values for m<0 can be obtained using equation (2.1). The
notation (3) (xxy[ indicates that there are three equal coefficients obtainable by
permuting the Cartesian suffixes. K=I to obtain Fano and Racah's phase
convention, or - i to obtain Condon and Shortley's.
We
Consider the vector T ~ Bobtained by contraction of the third-rank tensor T~0r.
have
T ~ = T~8~3~y
= ~Th,2%;p<lxlfls ;Pl ~
r 21' + 171/z
= XT,1,~3 ~ <ll ;Pl ~ >( - )'3-%~1 L2 & + 13
= Z < I ;Pl ~>V~{ - r101;~ +
lo
V3T111;,,- V5T131;,, }
or
( T. B~)I = ~ 1 {
_ TlOl + %/3 7"111- %/5 T m }.
(5.7)
Similarly
(TBa.)I = - V'3 riot.
(5.8)
1470
A.J. Stone
For TB~e we require
<lll~13;Pl fl~7'>3B~, 8t~,(12)<111213;P[ 1t~'>
= 3a~( - 1 )t*<lll2l3 ;p Io~fly>
=
and then equation (2.3) leads to
(Te.a)l = - VI{ Taol + V/3 Tlll + v ' 5 T m }.
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Now equations (5.7)-(5.9) can be solved for
(5.9)
T,I,, in terms of the traces :
T l o l = _ ~ / I ( T ~ B .)1,
T i l l = 1 ( T. B~)I -B)I'
T121= 89
- 3(T.aa)l + 2(T,,.)l - 3(T~.a)l},
89
l
(5.10)
and substitution into (5.3) yields finally
T[II1]t;p = - (15)-1/~( T. as + T~.~+ T ~ ]1;o
(5.11)
or in terms of transformation coefficients,
I [11111 ; p > = --(15)--1/2{ <0/1 1 ;p>a3y+ <311 ;p>8~v+ <~'l 1 ;p>3~o}.
(5.12)
Thus (5.12) and table 2 together provide a unitary transformation between a
symmetric rank 3 cartesian tensor and its spherical components.
6. FOURTH-RANK CARTESIAN TENSORS
The same procedure can be followed for the transformation coefficients of a
fourth-rank cartesian tensor. Let us confine our attention to tensors which
transform under permutation of suffixesaccording to the symmetric representation
of the group generated by (12), (34) and (13) (24)--i.e. which satisfy
A~=
A~=
A~y= A~.
(6.1)
This symmetry will be denoted by the symbol s. Symmetry with respect to all
permutations of suffixes is excluded by this notation, and is indicated by the
symbol S.
A programme written in PL/I by the author for performing exact arithmetic
on numbers representable as the square roots of rational fractions (including
calculation of 3 - j , 6 - j and 9 - j symbols, and elementary matrix manipulation)
was used on the IBM 370/165 computer at Cambridge to derive the following
results :
<a/3~,31 SO> = ] V 5 ( a/3y81 1010> + ~<afly31 1210>
= (1/3 V/5) [BoBBy~+ 3, ~as + ~, 83~r]
(6.2)
<afly3 ] s0> = -~(afly31 1010> + ~V'5 <afly31 1210>
= - ~[23~3 ~8 - 3~3t3, - 8~a3By]
(6.3)
<~/3~,~[ S2> = 10-'/211v'35 <~/3r3 [ 1012> + ~v/7 <~fl~,~11212> + C3 <~13~,311232>]
= - K~. 42-1'~[ (aft I 12 >8~, + <a~, I 12 >3r + <13112 >3B
+ <fl~112>8~+ <f13] 12>8~ + <~,3 [ 1 2 > 3 j
(6.4)
Cartesian and spherical tensors
1471
(a/39,31 s2 ) = - 1 ( ~/373 [ 1012 ) + ~ V'5 (~/39,31 1212 ) + 1~/3 (a/3y31 1222 )
+
] 12)8 o+ 112)so + 112) B
( r112)3 n- 2(oq~112)3v.- 2(r~112)3j.
T h e s e formulae can be evaluated with the help of table 1.
(6.5)
Finally we have
Downloaded by [North Carolina State University] at 05:56 03 September 2012
(o~fly~] $4 ; p ) = (oq3y3 [ 1234 ; p ) ;
these values are tabulated in table 3.
Common
factor
~XXXX
(4) (xxxy
(4) (xxxg
(6) (xxyy
(12) (xxyz
(6) (xxzz
(4) (xyyy
(12) (xyyz
(12) (xyzz
(4) (xzzz
(YYYY
(4) (yyyz
(6) (yyzz
(4) (yzzz
(6.6)
11234;0)
]1234;1)
]1234;2)
[1234;3)
[1234;4)
280 -1/2
3
224 -1/~
112 1/2
-2
-i
32-I/2
16-1/3
1
i
3
-1
i
-i
-1
1
-4
2
-i
1
2i
-4
3
2
3i
-4
-2
-4i
8
Table 3. Values of (ctfly3]1234; m) for m ~>0. The common factor multiplies all entries
in the column. Values for m < 0 can be obtained using equation (2.1).
REFERENCES
[1] CONDON,E. U., and SHORTLEY,G. H., 1935, The Theory of Atomic Spectra (Cambridge
University Press).
[2] FANO, U., and RACAH, G., 1959, Irreducible Tensorial Sets (Academic Press).
[3] STONE,A. J. (to be published).
[4] BRINK, D. M., and SATCHL~R, G. R., 1968, Angular Momentum (Oxford University
Press).
[5] YUTSlS, A. P., LEVlNSON, I. B., and VANAGAS,V. V., 1962, The Theory of Angular
Momentum (Israel Program for Scientific Translations, Jerusalem).
[6] BARRON,L. D., and BUCKINGHAM,A. D., 1974, y. Am. chem. Soc., 96, 4769.
[7] RECCE, T., 1959, Nuovo Cim., U, 116.
[8] HAMERM•SH, M., 1962, Group Theory (Addison-Wesley).